Helix of a curve

Definition

Let $\gamma$ be a parametrized curve on a manifold $M$. The helix or spiral of $\gamma$ is a parametrized curve in $\mathbb {R} \times M$ that sends $t$ in the parameter domain to $(t,f(t))$. In other words, the helix of a curve is the Cartesian product of the curve with the identity-parametrized curve.

Note that the above definition requires a parametrized curve. However, for a Riemannian manifold, any smooth curve in the manifold acquires a natural parametrization by arc-length, and we can thus give a natural choice of helix, or spiral, associated with the curve.

For a closed curve, viz a map $S^{1}\to M$, we construct the helix using the map $\mathbb {R} \to M$ that composes the modulo map with the curve parametrization.